Exam 6: Analytic Trigonometry

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\csc ^ { 2 } \frac { \theta } { 2 } - \sec ^ { 2 } \frac { \theta } { 2 } = ?$

Find the exact value of the expression. - $\frac { \tan 10 ^ { \circ } + \tan 20 ^ { \circ } } { 1 - \tan 10 ^ { \circ } \tan 20 ^ { \circ } }$

Solve the problem. -The function $I ( t ) = 40 \sin \left( 60 \pi t - \frac { \pi } { 2 } \right)$ represents the amperes of current produced by an electric generator as a function of time t, where t is measured in seconds. Find the smallest value of t for which the current is 20 amperes. Round your answer to three decimal Places, if necessary.

Simplify the expression. - $4 \sin 2 x \cos 2 x$

Simplify the expression. - $1 - 2 \sin ^ { 2 } 2 x$

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\sin \left( \tan ^ { - 1 } 2 \right)$

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\cos ^ { 2 } x - \sin ^ { 2 } x = 1 - 2 \sin ^ { 2 } x$

Find the exact value of the expression. - $\frac { \tan 70 ^ { \circ } - \tan \left( - 50 ^ { \circ } \right) } { 1 + \tan 70 ^ { \circ } \tan \left( - 50 ^ { \circ } \right) }$

Find the exact value of the expression. Do not use a calculator. - $\tan ^ { - 1 } \left[ \tan \left( \frac { 6 \pi } { 7 } \right) \right]$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\cos \left( \tan ^ { - 1 } \mathrm { v } \right) = ?$

Choose the one alternative that best completes the statement or answers the question. Express the product as a sum containing only sines or cosines. - $\sin ( 5 \theta ) \sin ( 2 \theta )$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\tan 75 ^ { \circ }$

Find the exact value of the expression. - $\cos ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right)$

Find the exact solution of the equation. - $\sin ^ { - 1 } x = \frac { \pi } { 2 }$

Find the exact value of the expression. - $\sin 25 ^ { \circ } \cos 95 ^ { \circ } + \cos 25 ^ { \circ } \sin 95 ^ { \circ }$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\sin \theta \cos ^ { 3 } \theta + \sin ^ { 3 } \theta \cos \theta =$ ?

Express the sum or difference as a product of sines and/or cosines. - $\cos \frac { 5 \theta } { 2 } + \cos \frac { 3 \theta } { 2 }$

Find the exact value of the expression. - $\frac { \tan 175 ^ { \circ } - \tan 55 ^ { \circ } } { 1 + \tan 175 ^ { \circ } \tan 55 ^ { \circ } }$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\cos 22.5 ^ { \circ }$

Find the exact value of the expression. - $\cos \left( \tan ^ { - 1 } \frac { 5 } { 12 } - \cos ^ { - 1 } \frac { 4 } { 5 } \right)$